1. Field of the Invention
The present invention relates to performing arithmetic operations on interval operands within a computer system. More specifically, the present invention relates to a method and an apparatus for finding the roots of a nonlinear equation of one variable using the interval version of Newton's method.
2. Related Art
Rapid advances in computing technology make it possible to perform trillions of computational operations each second. This tremendous computational speed makes it practical to perform computationally intensive tasks as diverse as predicting the weather and optimizing the design of an aircraft engine. Such computational tasks are typically performed using machine-representable floating-point numbers to approximate values of real numbers. (For example, see the Institute of Electrical and Electronics Engineers (IEEE) standard 754 for binary floating-point numbers.)
In spite of their limitations, floating-point numbers are generally used to perform most computational tasks.
One limitation is that machine-representable floating-point numbers have a fixed-size word length, which limits their accuracy. Note that a floating-point number is typically encoded using a 32, 64 or 128-bit binary number, which means that there are only 232, 264 or 2128 possible symbols that can be used to specify a floating-point number. Hence, most real number values can only be approximated with a corresponding floating-point number. This creates estimation errors that can be magnified through even a few computations, thereby adversely affecting the accuracy of a computation.
A related limitation is that floating-point numbers contain no information about their accuracy. Most measured data values include some amount of error that arises from the measurement process itself. This error can often be quantified as an accuracy parameter, which can subsequently be used to determine the accuracy of a computation. However, floating-point numbers are not designed to keep track of accuracy information, whether from input data measurement errors or machine rounding errors. Hence, it is not possible to determine the accuracy of a computation by merely examining the floating-point number that results from the computation.
Interval arithmetic has been developed to solve the above-described problems. Interval arithmetic represents numbers as intervals specified by a first (left) endpoint and a second (right) endpoint. For example, the interval [a, b], where a<b, is a closed, bounded subset of the real numbers, R, which includes a and b as well as all real numbers between a and b. Arithmetic operations on interval operands (interval arithmetic) are defined so that interval results always contain the entire set of possible values. The result is a mathematical system for rigorously bounding numerical errors from all sources, including measurement data errors, machine rounding errors and their interactions. (Note that the first endpoint normally contains the “infimum”, which is the largest number that is less than or equal to each of a given set of real numbers. Similarly, the second endpoint normally contains the “supremum”, which is the smallest number that is greater than or equal to each of the given set of real numbers.)
One commonly performed computational operation is to find the roots of a nonlinear equation using Newton's method. The interval version of Newton's method works in the following manner. From the mean value theorem,ƒ(x)−ƒ(x*)=(x−x*)ƒ′(ξ),where ξ is some generally unknown point between x and x*. If x* is a zero of ƒ, then ƒ(x*)=0 and, from the previous equation,x*=x−ƒ(x)/ƒ′(ξ).Let X be an interval containing both x and x*. Since ξ is between x and x*, it follows that ξ∈X Moreover, from basic properties of interval analysis it follows that ƒ′(ξ) ∈ƒ′(X). Hence, x * ∈N(x,X) whereN(x,X)=x−ƒ(x)/ƒ′(X).Temporarily assume 0 ∉ƒ′(X) so that N(x,X) is a finite interval. Since any zero of ƒ in X is also in N(x,X), the zero is in the intersection X ∩N(x,X). Using this fact, we define an algorithm for finding zero x*. Let X0 be an interval containing x*. For n=0, 1, 2, . . . , defineXn=m(Xn)N(xn,Xn)=xn−ƒ(xn)/ƒ′(Xn)Xn=1=Xn∩N(xn,Xn),wherein m(X) is the midpoint of the interval X. We call xn the point of expansion for the Newton method. It is not necessary to choose xn to be the midpoint of Xn. The only requirement is that xn∈Xn to assure that x*∈N(xn,Xn). However, it is convenient and efficient to choose xn=m(Xn).
Roots of an interval equation can be intervals rather than points when the equation contains non-degenerate interval constants or parameters. Suppose the interval version of Newton's method to find the roots of a single nonlinear equation has not yet satisfied the user-specified convergence tolerances. Then it is difficult to distinguish between the following three situations:                a) the current interval is a tight enclosure of a single interval root;        b) the current interval contains sufficiently distinct interval roots that they can be isolated with a reasonable amount of effort; and        c) the current interval contains point and/or interval roots that are so close to being indistinguishable that the effort to isolate them is unreasonably large with the existing wordlength.        
What is needed is a method and an apparatus for terminating the interval version of Newton's root finding method before iterations lose their practical value in isolating meaningfully distinct interval roots.